Given an elliptic curve of analytic rank at most one, Kolyvagin proved that the Tate-Shafarevich group is finite using his theory of Euler systems. In the higher rank cases, Kolyvagin gave a conjectural description of the structure of the Selmer group in terms of the Euler system. This conjecture has recently been proved by Wei Zhang for a large class of elliptic curves. We shall explain Kolyvagin's conjecture, discuss some of its remarkable consequences and overview the strategy of the proof. To put things in context, we shall begin with a brief survey on the current status of the BSD conjecture.
Let be an elliptic curve over a number field . Recall that the BSD conjecture relates the arithmetic invariants of with the an analytic object, the -function (see a lowbrow introduction).
There has been vast progress on the BSD conjecture in recent years. We give a very brief (highly incomplete) summary.
Analytic continuation. Due to the recent work of someone in the audience, we now know has analytic continuation for any totally real field of degree .
Rank. To make clear the rank part of the conjecture, we recall the -descent exact sequence The -Selmer group is a direct sum of several (denoted by ) copies of and a finite abelian -group. Conjecturally, is finite, and hence conjecturally for any . Consider the following implications for , The implication 1) is trivial. The implication 2) is due to Gross-Zagier and Kolyvagin (80s) for all . The implication 3) is due to Skinner-Urban (2000s) for good ordinary with surjective mod representation . Now consider the analogous statement for rank 1 case, The implication 1) is trivial. The implication 2) is again due to Gross-Zagier and Kolyvagin. The implication 3) is due to Wei Zhang (2013) for good ordinary with surjective under some mild ramification hypothesis.
BSD formula. In the case, the -part of the BSD formula is known for (under similar hypothesis) due to the work of Kato (2000s) and Skinner-Urban on the Iwasawa main conjecture for elliptic curves. In the case, the -part of the BSD formula is proved by Wei Zhang for under similar hypothesis.
We now brief review the Euler system of Heegner points to motivate the statement of Kolyvagin's conjecture. In the next section, we shall state Wei Zhang's result on Kolyvagin's conjecture and deduce some of its consequences mentioned above.
Let be an elliptic curve over of conductor . Let be an imaginary quadratic extension with discriminant coprime to (we will assume for simplicity). Let be the quadratic character associated to . We assume the Heegner condition: every prime factor of splits in . Then and the BSD conjecture predicts that is odd (in particular, ). How do you construct a point?
The key thing is that under the Heegner condition, we have an abundant supply of algebraic points on over the ring class fields of . Recall that classifies cyclic -isogenies between two elliptic curves . For , let be the order of of conductor . Under the Heegner condition, there exists an ideal with norm . Then we have a pair of elliptic curves with CM by , with kernel . This defines a point (a Heegner point) , which is defined over the ring class field of by the theory of complex multiplication. Here corresponds to the open compact subgroup of under class field theory (so is the Hilbert class field of ). Using the modular parametrization (mapping to 0) we obtain algebraic points . Taking the trace using the group law on , we construct a Heegner point .
The big theorem of Gross-Zagier we went through last semester is the following.
It follows from the Gross-Zagier formula that is infinite order if and only if . When is of infinite order, Kolyvagin used his theory of Euler system to prove the finiteness of . A simple version of his theorem looks like
The rough idea of Kolyvagin's proof is that he constructed a system of cohomology classes in derived from the Heegner points satisfying nice norm and congruence relations. The local information about these explicit cohomology classes were good enough to bound the Selmer group (via the local and global Tate duality and the Chebotarev density). More precisely, let be the sign of the functional equation, then it was actually shown that and (the latter is contributed by ).
We now briefly recall the construction of Kolyvagin's Euler system. The key thing is the following assumption:
The reason for this key condition is the following. Let Since is inert in , we have .
Then is indeed invariant under for . To check this, we need to show that lies in . Namely By assumption , so it remains to show that . This is true since by assumption : the trace of the Heegner point on is exactly the Hecke operator acting on , which projects to . The above relation is also the origin of the name Euler system: appears in the Euler factor of at .
Notice is nothing but the cohomology class of . In higher rank cases, the class is indeed trivial by Gross-Zagier. Kolyvagin's Euler system constructs more classes in . One can ask whether some of these classes could be nontrivial. An obvious but crucial observation is that this is the same as asking whether some is not infinitely divisible by .
Kolyvagin proved that in fact So increasing the number of prime factors of may help bring down the -divisibility! We define
The hope is that even in the case that is torsion, the cohomology classes eventually becomes nontrivial.
Assuming this key conjecture, Kolyvagin's work allows us to understand the refined structure of .
Starting from a nontrivial element by assumption , Kolyvagin constructed the whole -Selmer group (even in the higher rank case!). Moreover, its -eigenspace can be completely determined.
Now we are ready to state Wei Zhang's theorem precisely and then deduce some remarkable consequences.
Wei Zhang's proof beautifully blends various ideas and ingredients:
Selmer groups and the divisibility of Heegner points, 2013, www.math.columbia.edu/~wzhang/math/online/Kconj.pdf.
Kolyvagin's work on modular elliptic curves, $L$-functions and arithmetic (Durham, 1989), London Math. Soc. Lecture Note Ser., 153 Cambridge Univ. Press, Cambridge, 1991, 235--256.